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SAMSON: A Generalized Second-order Arnoldi Method for Reducing Multiple Source Linear Network with Susceptance SAMSON: A Generalized Second-order Arnoldi Method for Reducing Multiple Source Linear Network with Susceptance

Yiyu Shi, Hao Yu and Lei He EE Department, UCLA

Partially supported by NSF and UC MICRO Analog Devices, Intel an Partially supported by NSF and UC MICRO Analog Devices, Intel and d Mindspeed Mindspeed Address comments to lhe@ee.ucla.edu Address comments to lhe@ee.ucla.edu

Outline Outline

Review and Motivation SAMSON Algorithm Experimental Results Conclusions

Motivation for Model Order Reduction Motivation for Model Order Reduction

Deep submicron design needs to consider a large number of linear elements

Interconnect, Substrate, P/G grid, and Package

Accurate extraction leads to the explosion of data storage and runtime Need efficient macro-model

Nonlinear Elements

Linear Elements

Nonlinear Elements

Reduced Model

Non-RHS MOR vs. RHS MOR Non-RHS MOR vs. RHS MOR

• Non-RHS MOR reduces transfer

function H(s)

• First Order Method to handle L

PRIMA [Pileggi et al, TCAD’98]

• Second Order Methods to handle

S (susceptance)

ENOR [Sheehan, DAC’99] SMOR [Pileggi et al, ICCAD’02] SAPOR [Su et al, ICCAD’04] [Liu

et al, ASPDAC’05]

• Main Limitation
• Can only match up to ceil(n/np)

moments

• Accuracy is significantly limited

when the port number np is large

• RHS MOR reduces output vector y(s)

[Chiprout, ICCAD’04]

explicit moment matching is used

(lack in numerical stability)

EKS [J. M. Wang et al, DAC’00]

Implicit moment matching based on

Incremental orthonormalization

frequency domain shifting to deal with

1/s and 1/s2 terms

IEKS [Y. Lee et al, TCAD’05]

Based on the observation that there

are no 1/s and 1/s2 terms for PWL sources in finite time

Problems Still Remain Unsolved Problems Still Remain Unsolved

None of the existing RHS MOR methods can deal with RCS circuits with susceptance elements

Both EKS and IEKS are first order methods Directly applying them to RCS circuits cannot guarantee

passivity

There is still much room to improve accuracy

Incremental Orthonormalization causes error to accumulate When matching high order moments, it becomes inaccurate

None of the existing MOR methods can handle arbitrary independent inputs

Especially when they contain 1/si terms (i>0)

frequency domain shifting (inaccurate)

In the existence of infinite PWL sources

EKS and IEKS cannot consider s=0 (cannot perform DC analysis)

Major Contributions of SAMSON Major Contributions of SAMSON

It is an RHS MOR method

Compare with SAPOR and other non-RHS methods, it is more accurate Can handle a large number of ports

It can deal with all kinds of input sources accurately without frequency domain shifting or incremental orthonormalization

Numerically more stable, more efficient and more accurate in the whole

frequency domain, especially at DC (s=0)

It is based upon generalized second order Arnoldi method

Can handle RCS circuits with passivity

Outline Outline

Review and Motivation SAMSON Algorithm Experimental Results Conclusions

SAMSON Algorithm: Overview SAMSON Algorithm: Overview

Generalized Current Excitation Vector Generation

Augm ented System Transform ation System Linearization Second order state m atrices, incidence m atrices and arbitrary inputs Projected output vector Projection

Generalized Current Excitation Vector Generalized Current Excitation Vector

e

BJ s V s sC G = Γ + + ) ( ) (

∑ ∑

+ = =

Θ =

1

) (

m i n i i i i i

s s V s γ

Original system equation Superposed SIMO system equation Define the generalized current excitation vector Jex as Jex can be divided into two categories

Rational Irrational

Can be expanded into Taylor series

and take dominant terms. Then it becomes a special case of the Rational category

Multiply both sides by

e ex

BJ J =

m m n n ex

s b s b b s a s a a s J + + + + + + = ... ... ) (

1 1

∑

= −

=

n i m i i ex

s J s J ) (

m ns

b s b b + + + ...

1

SAMSON Algorithm: Overview SAMSON Algorithm: Overview

Generalized Current Excitation Vector Generation

Augm ented System Transform ation System Linearization Second order state m atrices, incidence m atrices and arbitrary inputs Projected output vector Projection

Augmented System Transformation Augmented System Transformation

To linearize, we have to make LHS exactly one order higher than RHS by raising the order of LHS by n-m .

Introduce auxiliary variables V1,

V2, … , Vn satisfying

Insert into superposed SIMO

system equation =>

∑ ∑

+ = =

Θ =

1

) (

m i n i i i i i

s s V s γ

Superposed SIMO system equation Augmented system equation

∑ ∑

+ = =

⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ Θ = Ψ

1 n i n i i i i i

s U s

T m n

V V V V U ] ,..., , , [

2 1 −

=

m n m n

sV V sV V sV V

− − −

= = =

1 2 1 1

..., , ,

∑ ∑

= − − + + =

Θ =

n i i i m n m n i m i i

s s V s

1

) ( γ

Here we assume m<n. The transformation for the n<m case can be derived similarly

SAMSON Algorithm: Overview SAMSON Algorithm: Overview

Generalized Current Excitation Vector Generation

Augm ented System Transform ation System Linearization Second order state m atrices, incidence m atrices and arbitrary inputs Projected output vector Projection

System Linearization System Linearization

Expand the augmented system at a given frequency Introduce auxiliary variables Z1, Z2, … Zn, satisfying and we can obtain

∑ ∑

+ = =

= ⇒

1

) (

n i n i i i i i

R U A σ σ σ

σ + = s s

1 1

) ( R Z U A A = − + σ σ

∑ ∑

+ = =

⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ Θ = Ψ

1 n i n i i i i i

s U s

T m n

V V V V U ] ,..., , , [

2 1 −

=

Augmented system equation

1 1 1 1 1

) ( ... ) ( R Z Z U A R Z Z U A R Z U A

n n n n n n n

= + − = + − = +

− − +

σ σ σ

... ...

1 1

+ = +

+ + n n n n

R U A σ σ

e.g., Insert it into

n n n n n n n n n

R Z U A R Z U A σ σ σ σ = + ⇒ = +

+ + + 1 1 1

and we get

... ... ) (

1 1

+ = + +

− − n n n n n

R Z U A σ σ

System Linearization System Linearization

Expand the augmented system at a given frequency Introduce auxiliary variables Z1, Z2, … Zn, satisfying and we can obtain

∑ ∑

+ = =

= ⇒

1

) (

n i n i i i i i

R U A σ σ σ

σ + = s s

1 1

) ( R Z U A A = − + σ σ

∑ ∑

+ = =

⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ Θ = Ψ

1 n i n i i i i i

s U s

T m n

V V V V U ] ,..., , , [

2 1 −

= ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − ) ( p q D V A I σ

Moment space equation Augmented system equation

1 1 1 1 1

) ( ... ) ( R Z Z U A R Z Z U A R Z U A

n n n n n n n

= + − = + − = +

− − +

σ σ σ

SAMSON Algorithm: Overview SAMSON Algorithm: Overview

Generalized Current Excitation Vector Generation

Augm ented System Transform ation System Linearization Second order state m atrices, incidence m atrices and arbitrary inputs Projected output vector Projection

Projection Projection

Find projection matrix using method similar to PRIMA But only take the first N rows (N is the number of the original variables) The moment calculation is efficient because the linearized system is sparse

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − ) ( p q D V A I σ

Q

ex

J s V s C s G ˆ ) ( ˆ ) ˆ ( = Γ + + ) )

Moment space equation Projection matrix Projected system equation Directly project on Jex instead of B

Outline Outline

Review and Motivation SAMSON Algorithm Experimental Results Conclusions

Experimental Setting Experimental Setting

Experiments are run on a PC with Intel Pentium IV 2.66G CPU and 1G RAM. All methods are implemented in MATLAB Time domain responses are calculated by IFFT (Inverse Fast Fourier Transformation with 1024 sampling points) The examples to be presented are from real industry applications (courtesy of Rio Design Automation)

Power planes and packages are modeled by RCS meshes On chip power/ground grids are modeled by RC meshes Vias and bumps are modeled by RC elements PWL sources are generated from SPICE characterization of

FPGA circuits

Moment Matching Moment Matching

With the increase of port number (from 1 to 20)

SAPOR (second order non-RHS) matches a decreasing number of moments SAMSON always matches the same number of moments.

5 10 15 20 5 10 15 20 25 (a) port number = 1 Order of moment Relative error SAPOR SAMSON 5 10 15 20 5 10 15 20 25 Order of moment Relative error (c) port number = 10 SAPOR SAMSON 5 10 15 20 5 10 15 20 25 (d) port number = 20 Order of moment Relative error SAPOR SAMSON 5 10 15 20 5 10 15 20 25 (b) port number = 5 Order of moment Relative error SAPOR SAMSON

Frequency Domain Accuracy Comparison Frequency Domain Accuracy Comparison

• (a) Frequency domain comparison between SAPOR, EKS, SAMSON and original with

attenuated sine waveforms . (b) Frequency domain comparison between SAMSON, Original, IEKS, EKS and SAPOR with PWL sources; All circuits are reduced to the same order.

• Only SAMSON is identical to the original
• EKS outperforms SAPOR due to the RHS MOR nature

(a) (b)

Time Domain Accuracy Comparison Time Domain Accuracy Comparison

• (a) Time domain comparison between SAPOR, EKS, SAMSON and original with attenuated sine

waveforms . (b) Time domain comparison between SAMSON, Original, IEKS, EKS and SAPOR with PWL sources; All circuits are reduced to the same order.

• Again only SAMSON is identical to the original
• EKS outperforms SAPOR due to the RHS MOR nature

(a) (b)

Scalability and Accuracy Scalability and Accuracy

Average time domain waveform error of SAMSON, EKS, IEKS and SAPOR with respect to the reduced order Different sizes of circuits from 200-70000 nodes are used. For each circuit 30% ports have independent PWL sources. SAMSON has the fastest waveform convergence

33X more accurate than EKS and IEKS at order 40 48 X more accurate than SAPOR The non-RHS method, SAPOR, does not converge

Runtime Comparison Runtime Comparison

Comparison of the reduction and simulation time under the same accuracy

• f up to 50 GHz on an RC mesh with 11,520 nodes and 800 ports

SAMSON runs the fastest

25X faster than direct simulation faster than EKS and IEKS but have a similar trend

Conclusions and Future work Conclusions and Future work

SAMSON is an RHS MOR method

Compare with SAPOR and other non-RHS methods, it is more accurate Can handle a large number of ports

SAMSON can deal with all kinds of input sources accurately without frequency domain shifting or incremental orthonormalization

Numerically more stable, more efficient and more accurate in the whole

frequency range, particularly at DC (s=0)

SAMSON is based upon generalized second order Arnoldi method

Can handle RCS circuits with passivity

Thank you!